Depth of field

Applet: Andrew Adams, Nora Willett
Text: Marc Levoy

In our first optics applet we introduced Gauss's ray
diagram, and in our second applet we considered the
relationship between the sizes of things in object space (in the scene) and
image space (on the sensor). This led us to talk about depth of field - the
topic of this applet.

Circle of confusion, depth of focus, and depth of field

When you move the sensor in a digital camera relative to the lens, the plane in
object space that is sharply focused moves as well. How far can you move the
sensor before a sharply focused feature goes out of focus? To answer this
question, we must first define what we mean by "out of focus". The standard
definition is that the width of the blurred image of the feature has become
larger on the sensor than some maximum allowable
circle of confusion. This size of this circle is arbitrary,
but a reasonable choice is to make it equal in diameter to the width of a pixel.
The thick black vertical bar at the right side of the applet above
represents one circle of confusion.

Once we've established this metric, it's easy to see that objects coming to a
focus too far in front of or behind this circle (too far to the left or right
of this bar on the applet) will have spread out to a size larger than the
circle when they strike the sensor. Now look at the dark red construction
lines drawn on the applet. These lines connect the edges of the lens aperture
(horizontal purple bars lying athwart the lens) to each side of the circle of
confusion. A moment's examination of the lines will convince you that "too
far" means outside the pink-shaded diamond lying astride the bar. The width of
this diamond, which is really a distance along the optical axis, is called the
depth of
focus for this optical arrangement. To avoid confusion we've drawn
the depth of focus in green on the snapshot at left.

Applying
Gauss's ray construction,
we can compute the position in object space that corresponds to the circle of
confusion in image space.
We've drawn this as a second, thinner vertical black bar on the
applet. We say that these two vertical bars are
conjugates.
Note that the height of the two bars are different. These heights are related
to one another by the lateral magnification of the lens, and can be computed
from one another using the light red lines that pass through the center of the
lens and strike the endpoints of the bars.

Repeating on the object side of the lens the same construction of dark red
lines we drew on the image side, we form another pink-shaded diamond. Scene
features inside this second diamond (in object space) will focus to positions
inside the first diamond (in image space). This means their blur will be no
greater than one circle of confusion, i.e. they will appear "in focus" on the
sensor. The width of the object space diamond is called the
depth of field.
For your convenience we've drawn the depth of field in green on the second
snapshot at left.

The depth of field formula

Looking at the dark red construction lines on the applet, it's clear that the
width of the pink-shaded diamonds will depend on the size of the circle of
confusion. Let's call the diameter of this circle C. It's also clear
that the width of these diamonds will depend on the distances they are from the
lens. Given the focal length of a lens and one of these two distances, we can
compute the other distance using Gauss's ray construction. Thus, we only need
two of these three variables. We'll use the focal length, denoted
f, and the distance to the in-focus plane in the scene (the center of
left diamond), denoted U. If you've been looking at the previous
applets, the latter distance, which is variously called focus setting,
focus distance, or subject distance, is the same as
so in previous applets.
Finally, you can see from the layout of the construction lines that the width
of these diamonds will depend on where these lines originate on the lens,
i.e. the diameter of the aperture. As we know from earlier applets, this size is specified by an F-number
N.

From these four quantities, and using algebra that captures the geometry of the
dark red construction lines, we can compute the width of the pink-shaded
diamonds and hence the depth of field. It is beyond the scope of this applet
to take you through this derivation, but you can find it in slides 43 through
47 in the
lecture titled "Optics I: lenses and apertures".
The final formula, which is only approximate, is

This formula is shown on the applet, along with the number we compute from it.
Below this is another number, labeled as "Depth of Field" on the applet. This
is the actual width of the pink-shaded diamond in object space, computed
analytically from the construction lines. The difference between these two
numbers highlights how much of an approximation the formula is at certain
scales. To make the construction lines easy to understand, we've set the
initial F-number to 0.5 and the initial circle of confusion to 20mm, but
neither setting is reasonable for a real camera. If you change the F-number to
2.0, you'll find that the numbers nearly match.

Playing with depth of field

At long last, let's play with the applet. Drag the circle of confusion slider
left and right. Notice the effect it has on depth of focus (on the right side
of the lens) and depth of field (on the left). As the circle gets bigger the
allowable blur size increases, and the range of depths we consider to be "in
sharp focus" increases. For small circles of confusion the relationship is
linear, as one would expect from the position of C in the depth of
field formula - in the numerator and not raised to any power. As the circle
gets very large the relationship becomes non-linear. At these sizes the
formula we've given isn't accurate anymore.

Now reset the applet and try dragging the F-number slider left and right. Note
that as the aperture closes down (larger F-numbers), the depth of field gets
larger. Note also that one side of the depth of field is larger than the
other. Beginning from the in-focus plane, more stuff is in good focus behind
it than in front of it (relative to the camera). This asymmetry in depth of
field is always true, regardless of lens settings, and it's something
photographers come to learn by heart (and take advantage of). Finally, note
that while N is not raised to any power in the depth of field formula,
the width of the diamond seems to change non-linearly with slider motion. The
reason for this is that for fixed focal length
f, aperture diameter A is reciprocally related to N
(through the formula N = f / A), and as the construction lines show,
the width of the diamond really depends on A.

Reset the applet again and drag the focal length slider. Note that the depth
of field changes dramatically with this slider, becoming especially large at
short focal lengths, which corresponds to wide-angle lenses. This dramatic
relationship arises from the fact that f appears in the denominator of
the formula, and it's squared. Formally, we say that depth of field varies
inversely quadratically with focal length. As photographers know, long
focal length lenses have very shallow depth of field.

Now leave the focal length slider at 50mm and start playing with the
subject distance slider. As the subject gets further away, the depth of field
increases. Once again note that the change becomes dramatic at long subject
distances. This arises from the fact that U appears squared in the
(numerator of the) formula. In other words, depth of field varies
quadratically with subject distance. Note also that for these settings of
C, N, and f, when the subject distance rises above
about 365mm, the far side of the depth of field (behind the in-focus plane
relate to the camera) becomes infinite, hence the computed depth of field
(called D.O.F. in the applet) says "Infinity".

The subject distance at which this happens for these lens settings is called
the hyperfocal distance.
Its derivation is given in slide 56 of the
lecture titled "Optics I: lenses and apertures".
The derivation also shows that the near side of the depth of field, i.e. the
rightmost tip of the pink-shaded diamond, is about half-way between the
in-focus plane and the lens. In the image at left, the hyperfocal distance is
indicated with a vertical green line, and the halfway point - the closest
distance that would be sharp, with a green dot. Photographers would say that
if they can figure out the hyperfocal distance for a particular focal length
and F-number, everything from half of that distance to infinity will be in
sharp focus. Wouldn't it be nice if you could press a button and the camera
would focus at its hyperfocal distance? Maybe someone can write such a plug-in
for future
programmable cameras (Pardon the shameless plug for our lab's
research.)

Confused by all these relationships? Don't worry - it takes even professional
photographers a long time to master them. If you can memorize the formula
you're ahead of most of them, because while it's not terribly complicated, you
won't find this formula in any how-to book on photography. To help you along,
the graph at left summarizes the relationships we've discussed. Click on the
thumbnail image to get a larger version. The graph contains the same four
camera settings we've considered:
C, N, and f, and U.
It also shows the side effects of changing these settings - something we
haven't talked much about. For example, changing the size of the circle of
confusion C (e.g. making it larger than a pixel) changes how large you
can print the image on photographic paper, or how large you can display it on
your screen, without it looking blurry.

Synthetic aperture photography

One normally associates shallow depth of field with a single-lens-reflex (SLR)
camera, because only they have a large enough aperture to create this effect.
However, if you allow yourself to capture, align, and combine multiple images,
then you can approximate this effect computationally. Here are several devices
we've built in our
research laboratory
that do this.

The most brute-force way to implement this idea is to build a large array of
cameras. Pictured at left is the
Stanford Multi-Camera Array. It's array of 100 webcams, aimed slightly
inwards so that they share a common field of view, and wired together so they
can be fired at once. If you capture a set of images using this array, place
the images so that one particular object lines up (i.e. falls in the same pixel
in all the images), and add the images together, you can approximate the depth
of field produced by a lens as large as the array. If the array is 15 inches
across like the one pictured at left, that's a very shallow depth of field!
Look at this
15-second video showing synthetic aperture focusing to see through foliage.
The input for this video was an array of 45 cameras spanning
a "synthetic aperture" 6 feet wide.

If you insert an array of microlenses into an ordinary camera, you can
individually record each of the light rays entering the camera, rather than
having groups of them brought to a focus inside pixels. The resulting dataset
is called a
light field.
We've done a lot of
research in our laboratory on microlens-based capture of light fields,
including building a camera whose images you can
refocus digitally
after you take the picture, as well as a
microscope
(pictured at left) with similar abilities. (The microlenses are inside the
red-circled housing.) Using this approach you can also change the depth of
field while you're digitally refocusing it.

The aperture on a cell phone camera is very small - almost a pinhole. As a
result cell phones have a large depth of field. However, if you record video
while moving the phone sideways, then you align and add the frames of video
together, you can simulate the large aperture of an SLR. If you own an iPhone,
we've written an app called
SynthCam - available in the
iTunes app store - that lets you do
this. In addition to their shallow depth of field, large apertures gather more
light. This is one reason SLRs take better pictures in low light than cell
phones. However, if you add many frames together, then you can match the
light-gathering ability of an SLR. Thus, pictures you take with this app will
be less noisy than pictures taken with the iPhone's native Camera app. If you
don't own an iPhone, here's a
web site with examples of photographs
created using the app.